Poisson nonnegative matrix-factorization model

Let $n,p,r$ be positive integers and write ${\mathbb{N}}_0=\{0,1,2,\dots \}$. An observed count matrix $X=(x_{ij})\in {\mathbb{N}}_0^{n\times p}$ is admissible when every row and every column contains a positive entry. A Poisson nonnegative matrix-factorization parameter of inner dimension $r$ consists of matrices

$W\in [0,\mathrm{\infty }{)}^{n\times r},H\in [0,\mathrm{\infty }{)}^{r\times p},{\lambda }_{ij}=(WH{)}_{ij}={\sum }_{k=1}^r{W}_{ik}{H}_{kj}\mathrm{.}$

At this parameter, the random counts $Y_{ij}$ are mutually independent and have Poisson means ${\lambda }_{ij}$. For $\lambda >0$, the Poisson law is

$\mathbb{P}(Y=m)=e^{-\lambda }\frac{{\lambda }^m}{m!},m\in {\mathbb{N}}_0;$

for $\lambda =0$, it assigns probability $1$ to $m=0$ and probability $0$ to every $m>0$. This model defines a likelihood for the observed matrix; it makes no convexity or global-optimality assertion about the factor parameters.

Hastie, Tibshirani, and Friedman, §14.6

Lee and Seung, §§2–3